Re: Optimizing the Four Convivial Ratios (Keenan Pepper tuning)

--------------------------------------------------
Optimizing the "Four Convivial Ratios"
A Footnote to Keenan Pepper's Noble Tuning
--------------------------------------------------

In a recent article on Keenan Pepper's Phi-based "Noble Tuning" where
the ratio between the chromatic semitone or apotome and the diatonic
semitone or limma is equal to Phi, ~1.61803398874989484820459, I noted
that one felicitous feature of this tuning is its close approximations
for the "Four Convivial Ratios" of neo-Gothic music.[1]

These four ratios, termed "convivial" because they occur in the same
general region of the spectrum of regular tunings, are 14:11 and 13:11
for regular major and minor thirds; and 21:17 and 17:14 for diminished
fourths and augmented seconds, available as "alternative thirds" in
these tunings.

While it is impossible to obtain all four of these ratios in any
regular tuning, Keenan Pepper's temperament with fifths at ~704.096
cents (~2.141 cents wider than 3:2) approximates all four ratios to
within 1.5 cents.

Having discussed this remarkable property of the tuning, I would like
to compare it with some other temperaments in the same portion of the
neo-Gothic spectrum.

Before looking at tables of ratios and interval sizes in cents,
however, it might be well to consider the musical significance of
those numbers, discussed in Section 1. Readers so inclined are invited
to turn directly to the matters of optimizations and comparisons
addressed in Section 2.

--------------------------------------------------------------
1. The Four Convivial Ratios: psychoacoustical interpretations
--------------------------------------------------------------

From one point of view, the choice of a set of ratios for regular and
alternative thirds such as 14:11, 13:11, 21:17, and 17:14 could be
seen as simply another arbitrary musical fashion. The use of large
integer ratios might be seen as a variation on the impressively large
ratios of medieval Pythagorean tuning, with large _primes_ as an
influence of 20th-century practice and theory from Kathleen
Schlesinger and Harry Partch to Ervin Wilson and LaMonte Young.

However, this still leaves the question: why _these_ specific ratios
as an intonational ideal for an esteemed variety of neo-Gothic regular
temperament?

Here there may be two psychoacoustical explanations, both of which may
be related not so much to the precise sizes of the four ratios as to
their general location on the continuum of intervals. Both of these
explanations may fit the general theme of "measured complexity."

----------------------------------------------------------
1.1. Regular thirds: Poised between Pythagorean and 17-tET
----------------------------------------------------------

Any inquiry into the tuning of major and minor thirds in neo-Gothic
music might begin with the role of these intervals in Gothic styles of
the 13th and 14th centuries in Western Europe based on Pythagorean
tuning with pure fifths and fourths.

In this Gothic tradition, the major third or ditone (81:64, ~407.820
cents) and minor third or semiditone (32:27, ~294.135 cents) are often
described as "imperfect concords," at once unstable and yet
_relatively_ blending. The rather complex Pythagorean ratios for these
intervals nicely fit their musical role, and Pythagorean tunings for
Gothic or neo-Gothic music are ideal in a wide variety of timbres.

As we increasingly temper the fifth in the wide direction, however,
the tension as well as complexity of these intervals gradually
increases to a point where their fit to a Gothic or conventional
neo-Gothic musical style can become problematic in many timbres.

By the point we reach 17-tone equal temperament or 17-tET (fifths at
~705.882 cents, ~3.927 cents wide), some rather careful choices of
timbre of the kind used by Ivor Darreg[2] and developed in practice
and theory by William Sethares[3] may be required in a neo-Gothic
setting.

At a size of ~423.529 cents, not far from 23:18 (~424.364 cents), the
relatively blending major thirds of Gothic and conventional neo-Gothic
styles may actually be heard as somewhat more tense or "dissonant"
than the relatively tense although somewhat "compatible" major
seconds. However, with Setharean timbral adjustments to mitigate the
tension between partials, these 17-tET thirds can have a very
stylistically apt quality of being at once relatively "concordant" but
unstable.

Since I took note of this quality of 17-tET in a paper on neo-Gothic
temperaments[4], David Keenan has offered a mathematical application
of a Phi-based mediant or "Noble Mediant" to certain interval ratios
in order to estimate the point of "equal gravitation" and maximal
complexity between two simpler ratios.[5] Note that this application
of Phi to integer ratios of intervals should not be confused with its
application to logarithmic fractions of the octave in scale
generation, as happens in Keenan Pepper's tuning.

Applying Dave Keenan's formula to find the region of maximal
complexity between the simpler ratios for major thirds of 4:5 and 7:9,
we find that it is indeed very close to 17-tET:

5 + 9 Phi
NobleMediant (4:5, 7:9) = --------- = ~422.487 cents
4 + 7 Phi

Whether from a viewpoint of pragmatic experience with tunings and
timbres, or from the mathematical model of Dave Keenan's "Noble
Mediant of complexity," we have a continuum of tension in the central
neo-Gothic region from Pythagorean to 17-tET. In Pythagorean tuning,
major thirds seem to fit the "relatively blending but unstable" ideal
over a wide range of timbres; by 17-tET, rather delicate timbral
adjustments may be in order to maintain the stylistic nuances of
"concord/discord."

Intuitively, the middle portion of this spectrum somewhere around
29-tET (fifths ~703.448 cents, ~1.493 cents wide) or 46-tET (fifths
~704.348 cents, ~2.393 cents wide) might be expected to have musical
qualities differing somewhat from Pythagorean and yet "milder" or
"gentler" than 17-tET, and stylistically fitting over a somewhat wider
variety of timbres.[6]

While this pragmatic or "Aristoxenian" explanation for the musical
appeal of the region by no means excludes other approaches including
the exploration of mediant and related relationships between large
integer ratios, it provides a ready aesthetic orientation to this
portion of the neo-Gothic spectrum.

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1.1.1. A heraldry of ratios: gradient intermediates
---------------------------------------------------

Along this engaging continuum of the central "neo-Gothic plateau"
between 5:4 and 9:7, large integer ratios serve as intellectually and
heraldically engaging placenames for subtle gradients of musical
color. The ratio of 14:11, for example, represents the Classic Mediant
between 5:4 and 9:7, (5+9):(4+7).

Likewise, the ratio of 13:11 for regular minor thirds represents the
Classic Mediant of 6:5 and 7:6, or (6+7):(5+6). Thus the ratio is
regarded as esteemed, but not as uniquely privileged: 32:27, 33:28, or
20:17 might have equally precious musical qualities in the neo-Gothic
realm.

The same kind of heraldry can apply to irrational ratios also, for
example the 29-tET major third at 10/29 octave or ~413.793 cents. As
it happens, this interval is almost identical to 127:100 (~413.794
cents).

A numerical "lineage" for this ratio derives it as a weighted
intermediate between the Pythagorean 81:64 and 23:18, the former
representing classic Gothic balance and the latter an interval near
the region of maximal complexity. Here the weighting is y/x=2/1 on the
side of 23:18, an interval itself derivable as an intermediate of 5:4
and 9:7 with the same weighting of y/x=2/1:

81 + 23 * 2 81 + 46 127
GradientIntermediate (81:64, 23:18) = ----------- = ------- = ---
64 + 18 * 2 64 + 36 100

Here it should be emphasized that this taking of mediants or
intermediates between already complex ratios is quite different from
the finding of Classic or Noble Mediants between intervals with simple
ratios tuneable by a "locking in" of partials. While the term
"Heraldic Intermediate" may nicely communicate the neo-medieval
qualities of this approach, the term "Gradient Intermediate" may more
specifically convey the idea -- and ideal -- of seeking fine
intonational shadings between other complex shadings.

While the idea of a region of the plateau intermediate in "complexity"
or "tension" between Pythagorean and 17-tET can explain the attraction
of ratios such as 14:11 and 13:11, the special charm of this region
becomes clearer when we consider also the "alternative thirds,"
augmented seconds and diminished fourth.

-------------------------------------------------------
1.2. Active polarity: Between simplicity and neutrality
-------------------------------------------------------

As is well known, in a medieval Pythagorean tuning diminished fourths
(eight fifths down) at 8192:6561 (~384.360 cents) and augmented
seconds (nine fifths up) at 19683;16384 (~317.595 cents) have ratios
almost identical to a pure 5:4 (~386.314 cents) and 6:5 (~315.641
cents), differing by only a schisma of 32805:32768 (~1.954 cents).
While these intervals occur incidentally in some conventional
14th-century music, they play an important role in the changing
musical and intonational styles during the era of roughly 1400-1450,
possibly starting somewhat earlier in certain localities such as late
14th-century Florence, for example.

In contrast, as we temper fifths more and more in the wide direction,
17-tET marks the point at which shrinking diminished fourths and
expanding augmented seconds momentarily converge into a single size of
"neutral third" at 5/17 octave, or ~352.941 cents, quite close to 11:9
(~347.408 cents).

About midway between these two conditions, we find a region from
around 29-tET to slightly beyond 46-tET where these "alternative
thirds" are at once active and yet somewhat "polarized," with the
diminished fourth as a kind of "submajor third," and the augmented
second as a "supraminor third." Or, if we prefer medieval terms, we
might describe these intervals as "subditonal" and "suprasemiditonal."

Along with regular thirds at or close to 14:11 and 13:11, these
"convivial" submajor and supraminor thirds at or near 21:17 (~365.825
cents) and 17:14 (~336.130 cents) contribute to the region's
distinctive musical charm.

Unlike the smooth Pythagorean schisma thirds -- a beguiling feature of
early 15th-century music -- these alternative thirds have a quality at
once active, and also somewhat "polarized" in comparison to the
neutral thirds of 17-tET.

From a systemic point of view, these thirds have a kinship to their
regular counterparts at or near 14:11 and 13:11; they have an active
and "passionate" quality inviting compelling cadential resolutions.
Thus 21:17 and 17:14 might be considered "convivial" to these regular
thirds in stylistic terms, as well as in sharing the same portion of
the neo-Gothic spectrum.[7]

To sum up, the "Four Convivial Ratios" may represent a two-dimensional
concept of "poised and measured complexity" between Pythagorean and
17-tET:

---------------------------------------------------------------------
Tuning Regular thirds Alternative thirds
---------------------------------------------------------------------
Pythagorean Relatively complex Smooth and "simple"
(81:64; 32:27) (~5:4; ~6:5)
---------------------------------------------------------------------
29-tET to 46-tET Somewhat more complex Active submajor/supraminor
(~14:11, ~13:11) (~21:17, ~17:14)
---------------------------------------------------------------------
17-tET Near-maximally complex Single neutral size
(~23:18, ~20:17) (~11:9)
=====================================================================

This discussion may have placed the "Four Convivial Ratios" in some
musical context. As may have already appeared, these ratios are by no
means exclusively favored in neo-Gothic music (covering a spectrum
ranging from Pythagorean to 22-tET or 27-tET), or even regarded
categorically as "first among equals," but rather as "esteemed among
equals."

------------------------------------------------------------------
2. Optimization Choices: Keenan Pepper's tuning and some neighbors
------------------------------------------------------------------

Having placed the Four Convivial Ratios in some musical perspective,
we can approach the problem of their optimization in regular
temperaments. Keenan Pepper's tuning nicely exemplifies a rough point
of balance between the four ratios, while other temperaments represent
various nuances of compromise leaning in one direction or another.

One way of getting an overview of the possibilities is a spectrum
chart for the range of temperaments with fifth sizes between 703.4
cents (just below 29-tET) to 704.7 cents (a bit beyond 46-tET or a
tuning with major thirds at precisely 14:11).

The top number line of the chart shows the spectrum of fifth sizes,
with eight tunings indicated: the tunings making each of the four
ratios "pure" (if this term can be applied to complex ratios); 29-tET
and 46-tET; Kennan Pepper's tuning (KP); and an "e-based" tuning (E)
in which the ratio of whole-tone to diatonic semitone or limma is
equal to Euler's _e_, ~2.71828182845904523536029.

An accompanying number line shows the tempering of fifths in the wide
direction from a pure 3:2, with the lines below showing sizes for the
four varieties of regular and "alternative" thirds and variances from
the respective ratios of 14:11, 13:11, 21:17, and 17:14, with the
locations of these ratios indicated by an asterisk (*) on a number
line:

17:14 14:11
29 13:11 KP 21:17 46 E
|----|----*----|----|----|----|*---|----|---*|---*|----|----|----|
703.4 .5 .6 .7 .8 .9 704.0 .1 .2 .3 .4 .5 .6 .7
Fifth size and tempering (3:2 = ~701.955)
|----|----|----|----|----|----|----|----|----|----|----|----|--
+1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7
Major third (14:11 = ~417.508)
-3.9 3.5 3.1 2.7 2.3 1.9 1.5 1.1 .7 .3 +.1 .5 .9 1.3
|----|----|----|----|----|----|----|----|----|---*|----|----|----|
413.6 414 .4 .8 415.2 .6 416.0 .4 .8 417.2 .6 418.0 .4 .8
Minor third (13:11 = ~289.210)
+.6 .3 0 -.3 .6 .9 -1.2 1.5 1.8 -2.1 2.4 2.7 -3.0 3.3
|----|----*----|----|----|----|----|----|----|----|----|----|----|
289.8 .5 .2 288.9 .6 .3 .0 287.7 .4 .1 286.8 .5 .2 285.9
Diminished fourth (21:17 = ~365.825)
+7.0 6.2 5.4 4.6 3.8 3.0 2.2 1.4 0.6 -0.2 -1.0 1.8 -2.6 -3.4
|----|----|----|----|----|----|----|----|---*|----|----|----|----|
372.8 .0 71.2 70.4 369.6 68.8 68.0 67.2 66.4 65.6 64.8 64.0 63.2 62.4
Augmented second (17:14 = ~336.130)
-5.5 -4.6 -3.7 -2.8 -1.9 -1.0 -0.1 +0.8 +1.7 +2.6 +3.5 +4.4 +5.3 +6.2
|----|----|----|----|----|----|*---|----|----|----|----|----|----|
330.6 31.5 32.4 33.3 34.2 35.1 36.0 36.9 37.8 38.7 39.6 40.5 41.4 42.3

As this diagram may suggest, at 29-tET we are approaching the point
where 13:11 becomes pure (fifth ~703.597 cents); just past 704 cents,
we reach the tuning for a pure 17:14 (~704.014 cents). Continuing
onward, we meet Keenan Pepper's Noble Tuning not quite halfway between
this point and the tuning for a pure 21:17 (~704.272 cents).

At around 704.348 cents, we meet 46-tET, almost identical to the
tuning with pure 14:11 major thirds (~704.377 cents). Thus this tuning
might be called the neo-Gothic counterpart to 31-tET, likewise nearly
identical to 1/4-comma meantone with pure 5:4 major thirds -- with the
distinction that while 14:11 and 5:4 are both esteemed ratios in these
respective stylistic settings, the latter is a distinct "valley" while
the former represents one of many shadings on the neo-Gothic plateau.

Slightly beyond 46-tET and the pure 14:11 tuning, with a fifth of
~704.607 cents, the e-based tuning features regular thirds leaning
slightly more toward 17-tET, and alternative thirds leaning slightly
more toward "neutrality." One curious trait is an augmented third
(e.g. Eb-G#, the Pythagorean "Wolf fourth") at ~550.676 cents, giving
an excellent approximation of 11:8 (~551.318 cents). This interval can
lend a unique flavor to certain neo-Gothic cadential progressions.[8]

To get another overview of the optimization problem, we can compare
variances for each of the four ratios as found in each of these
tunings:

---------------------------------------------------------------------
Tuning Fifth 14:11 13:11 21:17 17:14
(Tempering) (~417.508) (~289.210) (~365.825) (~336.130)
---------------------------------------------------------------------
29-tET ~703.448 ~413.793 ~289.655 ~372.414 ~331.034
(~+1.493) (~-3.715) (~+0.445) (~+6.588) (~-5.095)
---------------------------------------------------------------------
13:11 ~703.597 ~414.387 ~289.210 ~371.226 ~332.371
(~+1.642) (~-3.121) pure (~+5.400) (~-3.759)
---------------------------------------------------------------------
17:14 ~704.014 ~416.058 ~287.957 ~367.885 ~336.130
(~+2.059) (~-1.450) (~-1.253) (~+2.059) pure
---------------------------------------------------------------------
KP ~704.096 ~416.382 ~287.713 ~367.235 ~336.860
(~+2.141) (~-1.126) (-1.497) (~+1.410) (~+0.731)
---------------------------------------------------------------------
21:17 ~704.272 ~417.087 ~287.185 ~365.825 ~338.446
(~+2.317) (~-0.421) (~-2.025) pure (~+2.317)
---------------------------------------------------------------------
46-tET ~704.348 ~417.391 ~286.957 ~365.217 ~339.130
(~+2.393) (~-0.117) (~-2.253) (~-0.608) (~+3.001)
---------------------------------------------------------------------
14:11 ~704.377 ~417.508 ~286.869 ~364.984 ~339.393
(+2.242) pure (~-2.341) (~-0.841) (~+3.263)
---------------------------------------------------------------------
e-based ~704.607 ~418.428 ~286.179 ~363.145 ~341.462
(~+2.652) (~+0.920) (~-3.030) ~(-2.681) (~+5.333)
=====================================================================

If one wishes to approximate all four ratios within 1.5 cents, then
Keenan Pepper's tuning is an outstanding choice.

Both the above spectrum chart and this table may suggest how the size
of our alternative thirds is very sensitive to the amount of
tempering: a diminished fourth is formed from eight fourths up or
fifths down, while an augmented second is formed from nine fifths up
or fourths down.

Starting from Pythagorean, where these intervals form schisma thirds
very close to 5:4 and 6:5, we find that by 29-tET they have become
moderately complex at ~372.414 cents and ~331.014 cents. Their
variance from 5-based ratios is interestingly very close to that of
regular major and minor thirds in 12-tET, but in the opposite
directions, ~-13.900 cents and ~+15.393 cents respectively.

Curiously, the 29-tET augmented second nicely approximates the Classic
Mediant of 6:5 and 17:14, 23:19 (~330.761 cents), while the diminished
fourth is not too far from the Classic Mediant of 5:4 and 21:17, 26:21
(~369.747 cents). Thus these alternative thirds may share some of the
active "submajor/supraminor" flavor which becomes more clearly defined
as we move further into the "Four Convivial Ratios" region.

At a fifth size of around 704 cents or very slightly larger, the
locale of Keenan Pepper's Noble Mediant tuning, we are in the central
portion of the region offering close approximations of both 21:17 and
17:14. This local zone might be seen as extending roughly to 46-tET or
a pure 14:11 tuning.

By the point where we reach our e-based tuning at ~704.607 cents, the
diminished fourth has contracted to around 363.145 cents, while the
augmented second has expanded to ~341.462 cents. The contrast between
these two sizes of alternative thirds still has the "polarized"
quality of the region, but with a slight shading toward "neutrality."

If we continue to increase the tempering of the fifth, we will soon
reach the point at ~705.066 cents where the diminished fourth is at
19:16 (~359.472 cents), often considered a variety of neutral third;
the augmented second at ~345.594 cents would be very close to an
11:9. From here to 17-tET, we may be dealing with fine gradients of
neutrality.

To sum up, Keenan Pepper's noble tuning places us squarely in the
center of a region where both regular and alternative thirds
(diminished fourths and augmented seconds) have active qualities
lending themselves admirably to neo-Gothic music. At the same time,
the somewhat milder qualities of the regular thirds near 14:11 and
13:11 in comparison to 17-tET may make this tuning somewhat more
easily adaptable to a variety of timbres in "conventional" neo-Gothic
styles following the 13th-14th century concept of these intervals as
unstable but _relatively_ "concordant."

-----
Notes
-----

1. Tuning Digest [TD] 860:3 (Part 1) and 861:15 (Part 2), 3 October
2000.

http://www.egroups.com/message/tuning/13947
http://www.egroups.com/message/tuning/13971

2. See Brian McLaren, "A Brief History of Microtonality in the
Twentieth Century," _Xenharmonikon_ 17:57-110 (Spring 1998), at p. 81,
on Ivor Darreg's exploration of equal temperaments: "In building a
13-tone bar instrument, for example, Darreg deliberately used
wedge-shape bronze slats to 'tame' 13 with a unique mellow timbre." My
neo-Gothic response to the last quoted metaphor might be: "Don't try
to tame a tuning; seek an affable and mutual habituation."

3. William Sethares, _Relating Timbre and Tuning_,
http://eceserv0.ece.wisc.edu/~sethares/consemi.html

4. "Neo-Gothic Tunings and Temperaments," Part 1 (7 July 2000) and
Part 2 (8 July 2000). In Part 2, I wrote: "Much beyond Pythagorean, at
least when seeking to maintain a 13th-14th century balance of
concord/discord, we may 'Darregize/Sethareanize' our timbres to keep
our thirds _relatively_ blending while enjoying the 'superefficient'
resolutions they invite and the other special qualities of these
tunings. With the right timbre for 17-tET, a half-cadence in Machaut
on a major third can have the expected quality of a charming but
pregnant pause rather than an acute clash!"

http://www.egroups.com/message/tuning/11096
http://www.egroups.com/message/tuning/11108

5. See Margo Schulter and David Keenan, "The Golden Mediant: Complex
ratios and metastable intervals," TD 810:3 (17 September 2000).

http://www.egroups.com/message/tuning/12915

6. Based on my "unscientific" experience with some neo-Gothic
temperaments, I might tentatively conclude that the tuning of the
major third may be more sensitive than that of the minor third, which
might fit the Gothic role of an unstable "semi-concord" at almost any
value between the Pythagorean 32:27 (~294.135 cents) and 7:6 (~266.871
cents). Even in styles based on fully concordant thirds as the norm,
meantone augmented seconds in this same range have sometimes been
considered as acceptable consonances, and likewise 17-tET and 22-tET
sonorities with the minor third below and major third above (the
latter tuning very close to 6:7:9).

7. Another possible approach to explaining the musical "conviviality"
of ratios at or near 21:17 and 17:14 to neo-Gothic style is to note
that these intervals could be described as "inverse Pythagorean"
thirds respectively about a syntonic comma (81:80, ~21.51 cents)
_narrower_ than 5:4 and _wider_ than 6:5 respectively. The actual
variance in each case is 85:84 (~20.488 cents).

8. Using a MIDI-style notation with C4 as middle C, the resolution of
the sixth sonority B3-Eb4-G#4, with its "near-11:8" fourth between the
two upper voices, to A3-E4-A4 is a typical neo-Gothic progression
involving this interval. More specifically, the size of this "large
fourth," in comparison with a pure 11:8 (~551.318 cents), will be
~545.052 cents in Keenan Pepper's tuning; ~547.826 cents in 46-tET
(21/46 octave); ~548.147 cents in a regular tuning with pure 14:11
major thirds; and ~550.676 cents in the e-based tuning.

Most respectfully,

Margo Schulter
mschulter@value.net